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On Lucas pseudoprimes of the form $$ax^ 2+bxy+cy^ 2$$. (English) Zbl 0852.11006
Bergum, G. E. (ed.) et al., Applications of Fibonacci numbers. Volume 6: Proceedings of the sixth international research conference on Fibonacci numbers and their applications, Washington State University, Pullman, WA, USA, July 18-22, 1994. Dordrecht: Kluwer Academic Publishers. 409-421 (1996).
Lucas numbers $$\{L_k\}$$ are defined by the formula $$L_k= L\cdot L_{k-1}- M\cdot L_{k-2}$$ for $$k\geq 2$$, and $$L_0= 0$$, $$L_1 =1$$, where $$L$$, $$M$$ are rational integers, $$L>0$$. A Lucas pseudoprime is an odd composite integer $$n$$ coprime to $$D= L^2- 4M$$ and satisfying $$L_{n- (D/n)} \equiv 0\pmod n$$, where $$(D/n)$$ is the Jacobi symbol. Generalizing an earlier result [cf. the author and A. Schinzel, C. R. Acad. Sci., Paris 258, 3617-3620 (1964; Zbl 0117.28203)] the author proves that under some mild restrictions every primitive binary quadratic form $$ax^2+ bxy+ cz^2$$ represents infinitely many Lucas pseudoprimes.
For the entire collection see [Zbl 0836.00026].

##### MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11E16 General binary quadratic forms 11A15 Power residues, reciprocity